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THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


ELIMINATION 

BETWEEN 

TWO   UNKNOWN   EQUATIONS   WITH   TWO   UN- 
KNOWN   QUANTITIES, 

BY    MEANS    OF 

THE  GREATEST  COMMON  DIVISOR. 


ALSO, 


ANALYSIS   OP   CURVES, 


WITH    AN    APPLICATION     TO 


AN  EQUATION  OF  THE  FOURTH. DEGREE. 


rt" 


FRANCIS  H.  SMITH,  A.  M., 

Superintendent  and  Professor  of  Mathematics  in  the  Virginia 
Military  Institute. 


NEW-YORK: 
WILEY   AND   PUTNAM. 


1842. 


RESOLUTION,  &c. 


Resolution  of  two  equations  with  two  unknown  quantities. — 
Elimination  by  the  Greatest  Common  Divisor. 

1.  The  most  general  equation  of  the  mih.  degree,  be- 
tween two  unknown  quantities  x  and  y,  contains  all  the  terms 
in  which  the  sum  of  the  exponents  of  x  and  y  does  not  ex- 
ceed m.     Its  form  may  then  be  represented  by  the  equation 

3.^  +  Vx^-^  -f  Qa;"*-2  -I-  Ra;'"-3  .  .  .  . -\- Tx  +  u  =  0, 

in  which  P,  Q,  R,  &c.  are  functions  of  y,  as  follows  : 

P  represents  a  polynomial  of  the  first  degree  in  y  of 

the  form  a  +  by 
Q,  represents  a  polynomial  of  the  second  degree  in  y  of 

the  form  c  +  dy  +  ey^ 

R  represents  a  polynomial  of  the  third  degree  in  y  of 
the  form  f  ■\-  gy  -{- hy''  +  ly\ 
&c.    &c,    &c. 

the  last  co-efficient  u,  containing  all  the  powers  of  y,  from 

zero  to  m. 

2 


J^  JffiAn  equation  thus  formed  is  said  to  be  a  complete  equa- 
<r     '.f  the  mth  degree  between  two  unknown  quantities,  and 
..li  any  of  its  terms  are  wanting,  it  is  called  an  incomplete 
equation. 

3.  Could  we  solve  equations  of  every  degree,  the  ordinary 
methods  of  elimination  might  be  readily  applied,  to  the  solu- 
tion of  any  system  of  m  equations,  with  m  unknown  quanti- 
ties ;  and  we  should,  in  general,  obtain  a  determinate  num- 
ber of  solutions.  It  would  be  only  necessary  to  find  the 
value  of  one  of  the  unknown  quantities  in  terms  of  the  others, 
in  one  of  the  equations,  and  substitute  this  value  in  each  of 
the  other  equations ;  there  would  result  a  new  system  of 
equations,  with  one  less  equation  than  were  given,  and  with 
one  unknown  quantity  less.  By  continuing  this  operation, 
we  should  obtain  a  single  equation  with  but  one  unknown 
quantity.  This  equation  is  called  the  final  equation,  and 
serves  to  determine  the  values  of  the  unknown  quantity  which 
it  contains,  which  by  substitution  will  make  known  the  cor- 
responding values  of  the  others. 

4.  If  the  number  of  equations  exceeded  the  number  of  un- 
known quantities,  we  could  by  the  above  method  eliminate 
all  the  unknown  quantities,  and  there  would  result  one  or 
more  equations,  containing  only  known  terms,  which  would 
be  equations  of  condition  necessary  to  be  fulfilled,  in  order 
that  the  given  equations  should  not  be  incompatible  with 
each  other. 

5.  Should  the  number  of  unknown  quantities  exceed  the 
number  of  equations,  the  question  would  be  indeterminate  ; 
for  by  giving  arbitrary  values  to  as  many  of  the  unknown 
quantities  as  were  in  excess,  we  might  determine  the  values 


of  the  others  by  means  of  the  given  equations,  and  thus' 

as  many  different  solutions  as  there  were  arbitrary  ^liftaes' 

assumed. 

6.  But  the  difficulty  of  solving  equations  in  general,  has  led 
algebraists  to  seek  other  methods  of  elimination  than  the  one 
just  mentioned,  so  as  to  obtain  at  once  a  single  equation  in- 
volving but  one  unknown  quantity.  Various  methods  have 
been  used  to  determine  this  final  equation,  and  that  method 
is  regarded  as  best,  which  leads  to  a  final  equation,  whose 
roots  make  known  all  the  values  of  the  unknown  quantity 
which  it  contains,  which  are  compatible  with  the  given 
equations,  and  no  other  values.  The  method  by  the  Greatest 
Common  Divisor  is  not  free  from  the  objection  of  leading  to 
foreign  roots,  but  it  is  found  to  be  the  most  convenient  in 
practice.     We  propose  to  explain  this  method. 

7.  Let 

A  =  o  B  =  0 

be  two  equations  involving  x  and  y,  and  let  /3  be  any  assumed 
value  of  y.  If  we  substitute  this  value  in  the  place  of  y,  in 
the  given  equations,  there  will  result  two  equations, 

A'  =  0  B'  =  0 

which  contain  only  x  and  known  quantities. 

Now  it  is  evident  that  ^  can  only  satisfy  the  given  equa- 
tions, when  there  exists  at  least  one  value  of  x,  which  will 
reduce  the  two  quantities  A'  and  B'  to  zero  at  the  same  time, 
that  is,  satisfy  the  equations 


A'  =  0  B'  =  0. 

Let  a;  =  a  be  such  a  value  of  a:,  it  follows,  that  these  last 
equations  must  have  a  common  divisor  in  x^  since  they  will 
both  be  divisible  by  {x  —  a),  if  a  be  a  common  root.  When 
this  condition  is  fulfilled,  the  value  of  y  is  called  a  compati- 
hk  value.     Hence, 

Having  given  two  equations  with  two  unknown  quantities, 
a  value  attributed  to  one  of  the  unknown  quantities  will  he 
compatible,  when  its  substitution  in  the  given  equations  causes 
them  to  have  a  common  divisor,  which  is  a  function  of  the 
other  unknown  quantity. 

S.  The  above  principle  leads  directly  to  the  method  to  be 
pursued,  to  effect  the  resolution  of  the  given  equations.  For, 
since  every  compatible  value  of  one  of  the  unknown  quanti- 
ties, y,  for  example,  must  when  substituted  give  a  common 
divisor  in  x,  (if  the  two  equations  be  determinate,)  it  follows, 
that  if  the  proposed  equations  be  arranged  with  reference  to 
X,  and  we  seek  the  greatest  common  divisor  to  the  polyno- 
mials composing  them,  we  shall,  after  successive  divisions, 
find,  in  general,  a  remainder  which  contains  only  y  and 
known  quantities.  This  remainder  must  be  zero,  if  the 
given  equations  have  a  common  divisor  in  x.  Calling  R 
this  remainder,  we  shall  by  placing  it  equal  to  zero,  form  the 
equation 

R  =  0, 

which  is  the  final  equation  spoken  of.  This  equation  ex- 
presses the  condition  necessary  for  the  common  divisor  in  x 
to  exist.     The  roots  of  this  equation  substituted  in  the  given 


9 

equations,  will  cause  them  to  present  the  same  values  for  x, 
and  of  course  to  have  a  common  divisor  in  x,  if  no  foreign 
roots  have  been  introduced  in  the  process  of  finding  the 
greatest  common  divisor.  The  method  of  detecting  and  re- 
moving these  foreign  roots  will  presently  be  e:^amined. 

9.  It  will  generally  be  sufficient  to  substitute  the  values  of 
y  deduced  from  the  final  equation,  in  the  last  divisor  in  x,  in 
order  to  obtain  the  corresponding  values  of  x.  For,  if  we 
represent  by  Q,  the  quotient  resulting  from  the  division  of  A 
by  B,  and  by  R,  the  fi.st  lemainder,  we  shall  have 

A  =  B  X  Q  +  R. 

From  this  equation  it  is  evident,  that  if  any  values  of  x 
and  y  reduce  A  and  B  to  zero,  they  must  make  R  =  o  also. 
The  equations 

B  =  0  R  =  0 

will  therefore  make  known  the  values  of  x  and  y,  which  will 
satisfy  the  three  equations 

A  =  o  B  =  o  R  =  o. 

Reciprocally,  the  values  of  x  and  y,  which  satisfy  the  two 
last  equations,  will  also  satisfy  the  equation 


10 


The  determination  of  the  roots  of  the  given  equation  is 
then  reduced  to  that  of  the  equations 


^»^'    B  =  o  R  =  0. 


Dividing  now  B  by  R,  we  shall  obtain  an  equation 

B  =  R  X  Q'  +  R'. 
The  roots  of  the  equations 

B  =  0  R  =  o 

will  therefore  be  found  among  the  solutions  of  the  equations 

R  =  o  R'=o. 

And  if  R'  be  the  remainder  in  y, 

W  =  0 

will  be  the  final  equation,  the  roots  of  which  substituted  in 

R  =  0 

will  make  known  the  systems  of  values  which  correspond 
.to  the  equations 

R'  =  o        R  =  o        B  =  o        A  =  o 


11 


and  of  course  the  values  which  are  compatible  in  the  given 
equations. 


1 0.  Let  us  apply  the  foregoing  principles  to  the  following 
ample:  .^^J^ 


EXAMPLE  I. 

A  =  a;''  —  37/x'  +  {3y'  —  y  -{-  1)  x  —  y'  +  y'—2y  =  o 
B  =  x'  —  2yx  -\-  y^  —  y  =  o. 

Following  the  rule  for  obtaining  the  greatest  common  di- 
visor, we  have, 

First  Division. 

x*  —  3yx^  +  {3y'  —  y+l)x—y'  +  y^—2y'\x^  —  2yx-\-y'—y 
x^  —  2yx'^  +  {if  —  y)  X  \x  —  ?/  =  Q, 

—  yx'  +  {2f  +  \)x  —  f+y'  —  2y 

—  i/x"  +     2i/'x  —   y  +    y^ 

X  —  2y  =  R. 

Second  Division. 

X  —  2y 


X  —  2yx  +  y  —  y 
x^  —  2yx 


X  =  Q' 


f-v  =  R' 


In  order  that  x  —  2y  be  a  common  divisor  to  the  twa 


12 

given  equations,  the  last  remainder  must  be  zero.     We  have 
therefore  for  the  final  equation, 

♦  y"  —  y  =  o. 

The  roots  of  this  equation  are 

y  =  0  3/  =  1. 

Substituting  them  in  the  equation 

X  —  2y  ~  0, 

formed  by  placing  the  last  divisor  equal  to  zero  (9),  we  find 
the  corresponding  values  of^  to  be 

X  =  o  X  =  2, 

which  determine  the  solutions  of  the  given  equations. 

11.  Should  the  quotient  resulting  from  the  division  of  A 
by  B,  be  a  fraction,  the  denominator  of  which  contained  ei- 
ther or  both  of  the  unknown  quantities,  the  principle  devel- 
oped in  Art.  9.  would  no  longer  hold  good.  For  if  in  the 
equation 

A  =  B  X  Q  +  R, 


Q,  were  equal  to  — ,  K  containing  one  or  both  of  the  un- 


13 
known  quantities,  we  should  have 


The  values  of  a:  and  y,  which  reduce  A  and  B  to  zero, 

might  also  cause  K  to  be  zero.    -— -  would  then  become  -^ 

K  0 

and  might  have  a  finite  or  infinite  value      The  value  of  R 

would  also  be  finite  or  infinite,  and  could  not  in  either  case 

be  zero.     The  roots  of  the  given  equations  could  not  then 

be  found  from  the  solution  of  the  equations 


B  =  o  R  =  o. 

12.  To  avoid  fractional  quotients,  we  adopt  the  same  ex- 
pedients resorted  to,  in  obtaining  the  greatest  common  divi- 
sor, and  which  consist,  either  in  suppressing  the  common 
factors  of  the  dividend  or  divisor,  or  by  multiplying  by  some 
factor  which  will  render  the  division  possible.  We  shall 
thus  be  enabled  always  to  obtain  entire  quotients,  in  which 
case  the  method  of  Art.  9  may  be  followed. 

13.  In  seeking  the  greatest  common  divisor  to  two  poly- 
nomials, the  suppression  of  common  factors,  or  the  introduc- 
tion by  multiplication  of  a  new  factor,  does  not  affect  the  re- 
sult, but  it  is  not  the  case  in  the  solution  of  equations.  We 
shall  therefore  examine  the  consequences  resulting  from  the 
introduction  or  suppression  of  these  factors. 

14.  Let  us  take  the  equations 

A  =  a  B  =  o. 


14 

and  let  us  suppose  that  the  division  of  A  by  B  cannot  be 
effected  ;  which  supposes  that  the  co-efRcient  of  the  first 
term  of  B  contains  factors  of  y  which  are  not  common  to 
that  of  the  first  term  of  A.  Let  D  be  the  product  of  all 
these  factors,  and  suppose  D  common  to  all  the  terms  of  B. 
The  proposed  equations  will  take  the  form 


A  =  o  B'D  =  0. 

These  equations  may  be  satisfied  by  making 
A  =  o  B'  =  o. 

or 

A  =  o  D  =  0, 

We  might  then  suppress  the  common  factor  D,  provided  that 
to  the  solutions  of  the  equations 

A  =  o  B'  =  0, 

we  add  those  of 

A  =  o  D  =  o. 

To  obtain  the  solutions  of  the  last  equations,  we  find  the 
values  of  y  in  the  equation 

D  =  o, 


15 

and  substitute  them  in 

A  =  o; 

the  systems  of  values  of  x  and  y,  thus  obtained,  will  give  all 
the  solutions  belonging  to  the  given  equations. 

15.  Let  us  now  suppose  that  D  is  not  common  to  all  the 
terms  of  the  divisor,  and  that  an  entire  quotient  can  only  be 
obtained  by  multiplying  the  dividend  by  D.  We  shall  then 
have 

AD  =  o  B  =  0. 

These  equations  may  be  satisfied  by  either  of  the  systems 
of  equations 

A  =  6  B  =  0,         or         D  =  0  B  =  0. 

Hence  the  system  of  values  of  x  and  y  deduced  from  the 
equations 

D  =  0  B  =  0, 

must  be  suppressed  as  not  belonging  to  the  solutions  of  the 
given  equations,  if  they  are  found  existing  among  the  solu- 
tions of  the  final  equations. 


16 

16.  The  following  example  will  illustrate  the  case  alluded 
to  in  Art.  14  : 


EXAMPLE     ir. 

yx^  +  Zyx'  —  2/V  +  (y  +  1)  x  —  y  =  o  .  .  (1) 
^z"  +  (3?/  —  1)  a;^  —  y'x  +  3/  —  1  =  0  .  .  .   (2). 

The  result  of  the  first  division  gives  for  a  remainder 
x^  -\-  2x  —  y. 

Dividing  equation  (2)  by  this  remainder,  we  have  for  the 
second  remainder 

(y  —  1)  x-^  +  (y  —  1). 

This  remainder  having  the  common  factor  {y  —  1),  we 
suppress  this  factor  (Art.  14),  and  proceeding  in  the  opera- 
tion, we  have  for  the  last  divisor  {x  —  y),  and  for  the  final 
equation 

f  +  \=o. 

The  roots  of  this  equation  are 

J/  =  +■/  — 1  y  =  —^  — 1. 


17 


Substituting  these  values  in  the  equation  formed  by  put- 
ting the  last  divisor  equal  to  zero, 


X  —  y  =  0, 

we  have  for  the  corresponding  values  of  a; 

X  =  -f-/; 1  x  =  — V     |7 

Making  now^  the  suppressed  factor  {y  —  1)  equal  to  zero 
(Art.  14),  the  equation 

y  —  1  =  0, 

gives  y  =  1,  which  being  substituted  in  the  preceding  divisor 
placed  equal  to  zero,  viz  : 

x^  -\-  2x  —  y  ■=  0. 

The  values  of  a:  deduced  from  this  equation,  will  give  the 
solutions  to  the  given  equations  which  were  omitted  in  the 
suppression  of  the  common  factor  (?/  — 1). 

17.  For  an  application  of  Art.  15,  take  the  example: 

EXAMPLE    III. 

(y_l)a;»  +  2a:  — 5?/  +  3  =  o  ....  (1) 
t/a:"  +  Ox  —  lOy  =  0 (2). 


18 

To  render  the  division  possible,  we  multiply  the  polyno- 
mial in  equation  (1)  by  y,  and  going  through  the  operation  of 
division  we  have  a  remainder 

(—ly+Q)x  +  bf  —  ly. 

Since  we  have  introduced  a  factor  in  the  dividend,  we 
must  examine  whether  or  not  any  foreign  roots  have  been 
added  to  the  question.     Taking  the  equations  (Art.  15), 

y  =  o         yx^  -{-  9x  —  lOy  =  o, 
we  find  their  roots  to  be 

y  =  0  X  =  o. 

Should  the  final  equations  give  these  values  among  their 
solutions,  they  must  be  rejected,  as  not  belonging  to  the 
given  equations. 

Proceeding  now  to  the  second  division,  after  multiplying 
the  last  divisor  by  —  "7?/  +  9,  to  render  the  division  possible, 
we  obtain  for  the  last  divisor 

(—  7y  +  9)  z  +  5^  —  7y, 
and  for  the  final  equations 

25y'  —  70/  —  126y'  +  4Uy'  —  243y  =  o. 
The  roots  of  which  are  found  by  known  rules  to  be 


19 


y  =  o.y  =  Uy=S.y  =  ^l±JjL^,y  =  ^±:Lt^. 


The  corresponding  values  of  x  deduced  from  the  equation 

( — '^y  +  9)  X  +  53/"  —  ly  =  0, 

being 

x  =  o,  x  =  l,  x  =  2,  x  =  —  5  —  VJo,  x  =  —  5  +  VYo. 

Here  we  see  we  have  the  solutions  x  =  0,  y  —  0,  found 
above.  They  must  then  be  rejected,  and  as  the  multiplication 
in  the  second  division  by  —  7y  +  9  has  not  introduced  any 
foreign  roots,  the  proposed  equations  admit  of  the  four  fol- 
lowing solutions  : 


Cy^l 
1  \ 

^           —3  +  3^1^ 

sT          5 

(x=  1 

(x  =  —  b—  Vio 

(3^  =  3 

2^ 

(           —3  —  3  Vlo 

4^                5 

(x  =  2 

(  x  =  —  5+  Vio. 

If  we  had  substituted  the  roots  x  =  0,  y  =  o,  in  equatior? 
(1),  in  the  first  place,  we  should  have  seen  they  would  not 
satisfy  it,  and  we  might  have  concluded  that  they  formed 
no  part  of  the  solution  of  the  question. 


20 

18.  Examples  are  frequently  presented  in  which  it  is  ne- 
cessary in  the  process  for  finding  the  greatest  common  divi- 
sor, to  introduce  as  well  as  to  suppress  factors,  to  render  the 
division  possible.  The  foregoing  directions  must  be  ob- 
served, and  all  compatible  values  which  are  thus  sup- 
pressed must  be  joined  to  the  solutions  given  by  the  final 
equations  ;  while  those  which  have  been  introduced  must  be 
rejected. 

19.  If  the  given  equations  can  be  decomposed  into  com- 
mon factors,  their  resolution  will  be  very  much  simplified 
by  putting  these  common  factors  equal  to  zero,  separately. 
There  may  be  two  cases,  1st.  the  common  factor  may  be  a 
function  of  one  of  the  unknown  quantities  only ;  2ndv  It 
may  contain  both. 

20.  Let  us  examine  the  first  case.  Take  the  two  equations 

A  =  o  B  =  o, 

and  suppose  them  to  contain  a  common  factor  which  is  a 
function  of  a;^  we  may  substitute  for  the  equations  the  fol-- 
lowing : 

f(x)xF{x,y)  =  o  ....  (1) 

/(^)X?  {x,y)  =  o  ....  (2). 

These  equations  may  be  satisfied  by  making 

/(^)=o. 

As  this  equation  contEiins  only  x  and  known  terms,  it 


21 

will  give  a  determinate  number  of  values  of  x,  which  will 
satisfy  the  given  equations  independently  of  any  determi- 
nation of  y. 

But  the  given  equations  may  be  satisfied  by  either  of  the 
following  hypotheses,  viz  : 

/  (a:)  =  0  and  9  (a;,  ?/)  =  o  .  .  .  .  (3) 

/  (x)  =  0  and  F  (x,  y)  =  0  .  .  .  .  (4) 
or 

F  (x,  y)  =  0   and  9  (.t,  3/)  =  o  .  .  .  .  (5), 

But  the  solutions  resulting  from  the  systems  (3)  and  (4) 
do  not  differ  from  those  determined  by  the  equation 

/  {^)  =  0, 

for  the  values  of  x  resulting  from  this  equation,  will  when' 
substituted  in  the  equations 

cp  (x,  y)  =  o         F  {x,  y)  =  0, 

lead  to  a  determinate  number  of  values  of  y,  which  will  be 
included  in  the  solutions  of  equation 

/  (^)  =  o, 

the  roots  of  which  will  satisfy  the  given  equations  for  any 
values  of  y.     Hence,  to  determine  the  remaining  solutions 
of  the  question,  we  have  to  obtain  the  values  of  x  and  y  re- 
sulting from  the  condition  (5), 
4 


22 

F  (x,  y)  =  o  9  {x,  y)  =  o, 

the  values  of  which  we  can  determine  by  the  ordinary  rule. 

21.  Tf  now  the  common  factor  contain  both  x  and  y,  the 
two  equations  will  assume  the  form 

/  {x,  y)  xF  (x,  y)=o 
f{x,y)  X  (?{x,y)  =0. 

Making  in  the  first  place 

/  (^,  y)  =  o, 

the  two  equations  will  be  satisfied.  This  equation  shows 
(Art.  5,)  that  by  assuming  any  values  of  y,  we  shall  have  a 
determinate  number  of  values  of  x,  and  reciprocally,  the 
values  of  a:  being  assumed,  those  of  y  will  be  determined. 
The  equations  therefore  admit  of  an  indefinite  number  of  so- 
lutions, resulting  from  the  presence  of  the  common  factor, 

But  the  hypotheses  of 

/  {x,  y)  =  0     and  9  {x,  y)  —  o 
and 

/  {x,  y)  =  0     and  F  (x,  y)  =  0, 

will  also  satisfy  the  given  equations.  These  equations  can- 
not however  give  new  solutions  to  the  question,  since  they 


23 

will  necessarily  be  included  in  those  which  result  from  the 
condition 

/  {x,  y)  =  0. 

To  determine  new  solutions,  we  must  therefore  take  the 
final  conditions 

cp  {x,y)  =  o  F  {x,  y)  =  o, 

and  determine  the  values  of  x  and  y,  as  in  ordinary  cases. 

22.  To  apply  the  above  principles,  take  the  following 
example : 

EXAMPLE    IV. 
(a;2  ^  y2^   (yx  —  6)(x—l)=0 

{x^  +  y')  {2x  —  3y){x  —  y)=o. 

These  equations  give  an  indefinite  number  of  solutions 
(Art.  21)  in  consequence  of  the  common  factor  {x^  +  y^). 
Suppressing  this  factor,  we  have  the  two  equations 

(i/x  —  6){x  —  l)  =  o 
{2x  —  3y)  (x  —  y)  =  o. 

These  equations  are  satisfied  by  either  of  the  following 
systems  of  equations : 


24 

yx  —  6  =  0  2x  —  3y  =  0 

yx  — r  Q  =  0  X  —    y  =  0 

X  —  1=0  2x  —  32/  =  o 

X  —  1  =  0  X  —    y  =^  0, 

From  which  we  deduce  the  following  additional  solutions ; 

y  =  -f2  a:=  +  3 

y=—2  x  =  —  3 

y  =  ±v/6  x==±^Q 
y  =  +  i  a:=  +1 

y  =  +  1  a;  =,  4-  1^ 

23.        EXAMPLE    V. 

Let  us  take  the  equations 

y  {x  —  1)  {x  -}-  y)  X  {x  +  1)  {x^  —  2y  — 1)  =  o 
y  (^x—1)  {x  +y)  xy  {x"^  —  y^)  =  o. 

These  equations  have  three  common  factors,  viz :  y, 
X  —  1,   x  -\-  y. 

Placing  them  separately  equal  to  zero,  we  have  the  three 
equations, 

y  —  o         X  —  1=0         X  -[■  y  =  0, 


25 

The  first  result  shows  (Art.  20)  that  the  given  equations 
will  be  satisfied  by  a  value  of  ?/  =  o,  independently  of  any 
determination  of  x;  the  second  gives  a  root  of  x  =  \,y  be- 
ing indeterminate ;  while  the  third  shows  (Art.  21)  that  for 
the  common  factor  {x  +  y),  the  given  equations  admit  of  an 
indefinite  number  of  solutions. 

We  have  therefore  the  following  solutions  from  these  com- 
mon factors, 

\y  =  o  {x=l  I  x  =  —  y 

i  X  indeterminate     (  y  indeterminate     (  y  indeterminate. 

The  given  equations  present  in  addition  the  following  sys- 
tems of  equations : 

y  ^=  o  X  +  \  =  0 

y  =  o  x^  —  2y  —  l=o 

x'^  —  y^  =  0  X  -{-  1  =  0 

x^  —  y^  =  o  x^  —  2y  —  \  —  o, 

which  give  the  solutions 


1. 

y  =  0 

x  =  —  I 

2. 

y  =  0 

X  =  +    1 

3. 

y  =  o 

X   =  —  1 

4.  y  =  +  1  X  =  —  I 


26 

5.  y  =  — 1  ^  =  —  !• 

6.  y  =  1  +  v/2  a;  =  1  +   Vi 

7.  3/  =  1  +  ^2  ^=— 1  — v/2 

8.  ?/  =  1  —  V2  a;  =  1  —  V2 

9.  3/  =  1  —  ^2  ^  =  —  1  +  ^^2' 

Solutions  (1),  (2),  and  (3)  are  included  in  the  solution 
y  =  0     X  indeterminate, 

resulting  from  the  common  factor,  y,  while  solutions  (4),  (7), 
and  (9)  are  given  by  the  equation 

X  =  —y. 
We  have  therefore  the  following  new  solutions  only, 


y  =  _l       ,?/  =  !+  V2      (y  =  l—  ^2 
x  =  —  1      \  X  =\  ■\-  ■v/2      f  ^  =  1  —  ■\/2. 


24.       EXAMPLE    VI. 

ar^  +  (8y  —  13)  a:  +  3/2  —  7?/  +  12  =  0 
a^  — •  (43/  +  \)  X  -\-  y"^  ■\-  hy  =  0. 


27 

First  Division. 


x^  +  (8?/— 13)a;  +  ?/2  — 7y  +  12 


a:^ —  {\y  +  1)  ar  +  3/^  +5^ 


(12y  —  12)  :c—  122/  +  12. 
This  remainder  can  be  decomposed  into  factors,  as  follows  : 

12(y— l)(a;— 1). 

The  question  is  thus  reduced  to  the  soKition  of  the  fol- 
lowing system  of  equations : 

1.   ^3/-!=" 

Lr2—  (4?/  +  1)  a;  +  y2  ^  5r/  =  o 


2. 


X  —  1=0 

ap-  —  (4?/  +  1)  a:  +  ?/^  +  5y-=  o- 


The  solutions  of  which  are  readily  found  to  be 

2/  =  1  ar  =  3 

y  =  1  a:  =  2 

y  =  0  a:  =  1 

y^—\  x= I 

25.  When  the  final  equation  is  independent  of  y,  and  con- 
tains known  terms  only,  which  do  not  of  themselves  reduce 
to  zero,  the  given  equations  are  contradictory,  and  cannot  be 


28 

satisfied  by  the  same  values  of  x  and  y  ;  for,  the  condition' 
of  their  having  common  values,  requires  that   they  should 
have  a  common  divisor  (iVrt.  7),  and  this  common  divisor 
cannot  exist  where  the  final  equation  is  not  satisfied. 
The  following  example  will  illustrate  this  principle. 


EXAMPLE    Vir. 


x^  —  i/  +  3  =  0. 


First  Division. 


yx^  —  {y^—3y  —  l)x  +  y 
yx'  —  (/  —  3y)  X 

X  +  y. 


Second  Division. 


x^  _  y2  _^  3' 


yx 


T^  —  3/^  +  3       ^  +  y 
X  +xy  x  —  y 

—  xy  —  y^  +3 

—  xy  —  y^ 

+"3. 


The  last  remainder  being  3,  the  final  equation  is 


But  this  equation  is  absurd,  since  3  cannot  be  equal  to 
zero.  The  proposed  equations  have  therefore  no  common 
divisor,  and  are  consequently  contradictory. 


29 

26.  Should  the  final  equation  reduce  to  zero,  of  itself,  the 
given  equations  will  contain  a  common  divisor  independently 
of  any  determination  of  y.  If  this  common  divisor  contain 
only  one  of  the  unknown  quantities,  x  for  example,  the  equa- 
tions would  be  satisfied  by  a  definite  number  of  values  of  :r, 
y  being  indeterminate  (Art.  20)  ;  while  they  would  admit  of 
an  infinite  number  of  solutions  if  it  contained  both  x  and  y 
(Art.  21).     Take  the  following  equations  : 


EXAMPLE    Vlir. 

x""  —  2yx'^  +  2y'^x  —  5.r-  +\Oyx  +  6.c  —y''  —  rnf  —  Qy  =  o 
x^  —  byx"^  +  Sy'^x  —  x  —  4?/^  -\-  y  =  o. 

After  the  third  division  we  obtain  for  the  common  di- 
visor, there  being  no  remainder, 

(y  *— 1 0^'+35/_50 // + 2i)x--z/  + 1 0y'—35y'-{-50y^—24y. 

The  proposed  equations  have  therefore  a  common  factor, 
and  by  putting  the  above  common  divisor  equal  to  zero,  we 
have 

X  =  y. 

Hence  x  —  y  is  a  common  factor  to  the  two  equations, 
and  they  therefore  admit  of  an  indefinite  number  of  solutions 
(Art.  21.) 

If  we  divide  the  given  equations  by  this  common  factor, 
we  shall  have  the  two  equations, 
5 


30 

x^  —  {2y -\- h)  X  -{- y"^  +  by -\- Q  =  0 
x^  —  Ayx  +  Ay^  —  1=0. 

Operating  upon  these  equations  by  the  ordinary  rule,  we 
shall  have  for  the  last  divisor  in  x, 

(2y  _  5)  a;  +  5y  —  3y2  +  7, 

and  for  the  final  equation, 

y*  —  iQrf  -}-  35?/2  _  50?/  +  24  =  o ; 

from  which  we  deduce  the  following  solutions, 

y  =  1  a:  =  3 

y  =  2  .-c  =  5 

y  =  3  X  =  5 

y  =  4  X  =  7. 

27.  The  substitution  of  the  values  of  y  deduced  from  the 
final  equation  in  the  divisor  of  the  first  degree  in  x,  may 
cause  the  values  of  x  to  assume  either  of  the  following 
forms,  viz : 


(1)  X  =  a,{2)x=  0,  (3)  X  =  6,{4)x  =  -. 


28.  Represent  the  divisor  in  x  by  Aa:  —  B,  A  and  B  be- 


31 

ing  functions  of  y.  In  the  first  case,  if  3/  =  B  be  the  value  of 
y,  which  by  substitution  in  the  equation 

kx  —  B  =0, 

gives  x  =  <x,  the  given  equations  admit  of  but  this  value  x, 
corresponding  to  the  value  of  y  =  /3  ;  since  the  equation 
from  which  the  value  of  a:  is  obtained  is  of  thej^rs^  degree 
only,  and  can  give  but  one  solution.  This  is  also  the  case 
when  the  value  of  y  gives  a;  =  0. 

29.  In  the  third  case,  when  we  find  x  =  ^,  for  the  value 
of  y  =  /3,  the  two  equations  are  contradictory ;  for  the 
equation 

Kx  — .  B  =  0 

can  only  give  x  =^,  when  the  substitution  of  the  value  of 
y  makes  A  =  0  and  B  equal  to  a  finite  quantity. 

But  when  A  =  0,  we  have  from  the  nature  of  the  above 
equation,  B  =  o  also.     Hence  the  equation 

kx  —  B  =  0 

is  absurd,  for  a:  =  ^  and  y  =  /?.  Further,  the  number  B  is 
the  common  divisor  which  the  substitution  of  y  =  /3  causes 
the  given  equations  to  acquire,  and  if  all  the  values  of  y  pro- 
duce in  the  same  manner  a  numerical  common  divisor,  it  is 
evident  no  values  of  x  and  y  can  satisfy  the  conditions  of  the 
questions.  The  proposed  equations  are  therefore  contra- 
dictory. 


32 

30.  Finally,  if  y  =  [3  reduce  A  and  B  to  zero,  at  the  same 

time,  the  value  of  x  becomes  -  or  indeterminate. 

o 

This  result  shows  that  the  equation  formed  by  placing  the 
divisor  of  the  first  degree  equal  to  zero,  does  not  make  known 
all  the  values  of  a:,  which  will  satisfy  the  proposed  equations 
for  the  value  of  y  =  13,  since  this  equation  reduces  to  zero,  by 
the  substitution  of  this  value  of  y,  independently  of  any  deter- 
mination of  X.  It  is  therefore  indeterminate,  and  if  the  value 
of  y  be  substituted  in  the  given  equations,  it  will  cause  them 
to  have  a  common  divisor  in  x,  of  a  higher  degree  than  the 
first.  The  degree  of  this  divisor  will  depend  upon  the  number 
of  multiple  values  of  x,  which  correspond  to  the  same  value 
of  y.  It  will  be  of  the  second  degree  if  there  be  two  values 
of  a;  to  one  ofy,  of  the  third,  if  three,  &c.  When  therefore 
the  divisor  in  the  first  degree  becomes  indeterminate,  by  the 
substitution  of  a  value  of  y,  deduced  from  the  final  equation, 
we  make  the  substitution  in  the  next  superior  divisor.  If 
this  divisor  be  of  the  second  degree  in  x,  and  admit  of  solu- 
tion, there  will  be  ticn  values  of  a:  corresponding  to  one  value 
ofy.  If  this  equation  also  be  indeterminate,  we  proceed  to 
the  next  superior  divisor,  and,  in  general,  to  that  divisor 
which  does  not  reduce  to  an  indeterminate  form. 

31.  If  we  knew  apn'on*,  from  the  composition  of  the  given 
equations,  that  they  contained  multiple  values  of  .r,  for  the 
same  value  of  y,  we  might  at  once  substitute  the  value  of  y 
in  the  divisor  of  the  degree  corresponding  to  the  number  of 
multiple  roots  ;  since  its  substitution  in  a  divisor  of  an  infe- 
rior degree  would  lead  to  an  indeterminate  result. 

32.  If  all  the  values  of  y  gave  multiple  values  of  x,  the 
operation  for  obtaining  the  greatest  common  divisor  would 


33 


necessarily  stop  at  a  divisor  of  a  degree,  corresponding  to 
the  number  of  tliese  multiple  roots  ;  as  is  shown  by  the  fol- 
lowing example. 


EXAMPLE    IX. 

x'  +  2yx'  +  (2.y2  +  1 )  o;^  +  (y^  +  Qy^  +  y  ^  8 1 )  a:  +  2/'  =  o 
x'  +  2yx2  +  2y'^x  +  y'  +  9?/2  —  81  =  o. 

First  Division. 


2-^+27/x3+(-2(/2-fl)c3+(2/3+9,/2-f7y— 81).r+3/2 


2:3+2y.r24-2y2.T+y3-f  9^/2—81 


x'+y-i:-\-y^ 


Second  Division, 


x^  +  2yx^  +  2y'^x  -\- if  +  9y^  —  81 
x^  +    y:?:^  -j-    y-x 

x^  -\-yx-{-  y^ 
X  +y 

yx'^  +  ?/^x  +  y^ 
yx^  +  ylr  +  y^ 

~^f  —  ^^- 

In  this  example  the  operation  stops  at  a  divisor  of  the  se- 
cond degree  in  x,  the  final  equation  being 

9/  —  81=0, 


so  that  for  each  of  the  values  of  y  -  ±  3,  deduced  from  this 
equation,  there  are  two  values  of  x. 


34 

33.   The  degree  of  the  final  cannot  exceed  the  product  of  the 
numbers  which  represent  the  degrees  of  the  given  equations. 

M.  PoissoN  demonstrates  this  principle   in  the  following 
manner: 

Let 

a:"'  +  Px—^  +  Qa:'"-2  + .  .  .  +  Tx  +  m  =  o 
a;"  +  Fa:"-'  +  Ofx"'^  +  .  .  .  + Tx  +  u'  =o, 

be  the  two  given  equations  ;  the  coefficients  P,  Q,  P',  Q',  &c. 
being  functions  of  y  of  the  most  general  form  (Art.  1),  as 
follows  : 

V  =  a  +  hy,V'  =  a'  -\-  b'y,  Q  =  c  +  ^y  +  ey\  &c.  «Sz;c. 

If  we  substitute  for  P,  P',  &c.,  their  values,  the  above 
equations  become 

x-^+ia^  by)  x^-'  +  {c  +  dy  +  ey^  3:^-2  .  .  .  ^  y^  =  0 
a:"  +  {a'  -f  b'y)  a:"-*  +  (c'  +  dly  +  e^)  a:"-^    .  .  .  +  y"  ^-r  0. 

But  the  degree  of  the  final  equation  will  not  be  dimin- 
ished, if  we  reduce  the  coefficients  of  these  last  equations  to 
the  term  which  contains  the  highest  power  of  y,  since  the 
degree  of  the  two  equations  will  not  be  changed  by  this 
operation.    We  shall  then  have 

a;"'  +  it/a;""-'  +  ey^x'"'^  .  .  .  +  2/"'  =  0 


35 

x"  +  b'yx"-'  +  e'y^x'"-^  .  .  .  +  y"  =  o; 

which  may  be  placed  under  the  form 

If  we  regard  I -I  as  the  unknown  quantity  in  these  equa- 
tions, and  represent  by  a,  [3,  y,  &c.  the  roots  of  i  -  I    in   the 

first,  and  by  a',  ^',  /',  &c.,  those  in  the  second  equation,  we 
shall  have 


or 


(a;  —  ay)  (x  —  (3y)  {x  —  yy)  &c.  =  o  .  .  (1) 
{x  —  ci'y)  {x  —  (S'y)  {x  —yy)  &c.  =  o  .  .  (2). 


36 

If  now  we  substitute  in  equation  (1)  each  of  the  roots  of  a: 
deduced  from  equation  (2),  viz : 

x  =  a.'y         X  —  /S'y         X  —  y'y,  &c., 
we  shall  have  n  equation  of  the  following  form, 

rj"  (a'  _  a')  (a'  —  /3)  (a'  —  y)  &C.  =  0 
y-{^'~a)  (/3'_^)  (/3'  — 7)&c.  =  o 
3/"  (7'  —  a)    (j'  —  (5)    (7'  —  7)  &C.  ~  0,  &LC.  &C. 

each  being  of  the  mi\\  degree,  and  giving  m  values  of  y. 
The  whole  number  of  values  of  y  will  therefore  be  m  x  n, 
which  will  represent  the  degree  of  the  final  equation.  The 
degree^  of  the  final  equation  cannot  therefore  exceed  this 
number. 

EXAMPLES. 


1. 


yx  —  y^  —  y  —  1  =  0 
yx  — ■y'^  —  1  =  0. 


Final  equation,  —  y  —  0.     Common  divisor,  yx  —  t/^  —  1. 
Equations  contradictory.    (See  Art.  29.) 


2.   ..2 


x^+  (8y— 13)  a; +  7/2 —  73/ -I- 12  =  0 
(4?/+    l)a;  +  ?/2  + 5?/  =  o. 


Common  divisor,  ~  [y  —  1)  (12x  —  12).     Final  equation, 
{y~\){y^  +  y)^o. 


37 

Solutions. 
y  =  \,  2/  =  l,  y  =  o,  y  =  —i 

0^  ^^  ^f    CC  ^^  Oj    2^  =—  Ij   CC  ^^  L» 

I  x'—3yx'-\-3x''+3xf—6yx—z--y^+Si/+i/—3=o 
^'   \  x'+3yx'—3x'+3xy'—6yx—x+y'—3y'—y+3=o. 

The  remainder  of  the  second  degree  in  x  is  divisible  by 
(y' —  1),  which  we  suppress  ;  after  this  suppression,  the  re- 
mainder of  the  first  degree  is  divisible  by  y^ —  2y.  Final 
equation  then  becomes  y^  —  2y —  3  =  o,  and  the  common  di- 
visor x.     (See  Article  14.) 

Solutions. 
y=^l,y=l,y=l,  y=o,  y=o,y  =—  l,y  =  2,y  =  2,  y  =  S 

x  =  o,  X  =  2,  x  =  —  2,  x—1,  x  =  — 1,  x=o,x—l,  x= — 1,  x  =  o. 

J  3a;2  —  5ya;2  —  {3y^  —  30y)  x  +  30y^  =  o 
(  6z2_  102/2  +  Uxy=o. 

Final  equation  in  x,  113a:^  —  1310:i-'  +  1800:r'  =  o. 

Solutions. 

a:  =  o,  X  =  0,  a;  =  10,  a;  =  — - 

1  1  O 

72 
y  =  o,  y  =0,  y  =  15,  y  =  — _^ 


38 


x^  +  y  —  5  =  0 
x^  -\-  f  a-y  ■\-  y^  =  o. 


Solutions. 
y=2,  y=—  2,  y=  1,  y  =  — 1 
x  =  —  1,   X  =  I,   x  =  —  2,  a:  =2 


6. 


(  x^  +  2xy  -\-y^  —  1  =  o 
(  x^ —    y^ — Qy  —  9  =  0. 

Solutions. 

y  =  —  \,         y  =  —2 
a;  =       2,  X  =       1. 


„    ^  x^  —  2yx  -\-y^  —  1  =  o 

i  x"" +2{y  —  5)x-\-y~—\(iy  +  2\=n. 


These  equations  can  be  placed  under  tlie  following  form 

\x-{i,+  \)\    \x-(y-\)\=^o 
\x-{^  —  y)\    \x  —  {l  —  y)\=o. 

Solutions. 
y  =  l,  3/  =  3,  3/ =  2,  y  =  4 

x  =  2,  x=4,  x=l,  x  =  3. 


3d 

?  a;2— 33/a;  —  5y  +  2y2_  lly  _  6  =  o. 
These  equations  may  be  written  thus  : 

\x  —  {y  +  l)\        \x—{3  +  y)\=o 
la:  — (2y— 1)^    \x  —  {6-^y)\=o. 
Putting  these  factors  two  and  two  equal  to  zero,  we  have 
x  =  3,  z  =  7,  1  =  6,  3  =  6, 

y  =  2,  y  =  4. 

The  two  last  results  are   absurd.     Final  equation,  y^  — 
6y  -{-  8  =  0. 


x^  —  4yx  +  4y'  —  1  =  o 


Which  may  be  placed  under  the  following  form 
\x  —  {2i/+l)\   \x  —  {2y—l)\=o 
)x  —  (2y  +  3)(   \x  —  {2y  +  2)\=oi 

which  furnish  the  following  absurd  results  : 


40 

1  =  3,     1=2,      —1=3,      —1=2. 

If  we  had  applied  the  ordinary  rule  for  determining  the 
final  equation,  we  should  have  found  a  numerical  remain- 
jdei  (see  Art.  25.) 

Suppressing  the  common  factor  {x  —y),  Art.   14,  we  have 
the  following  systems  of  equations : 

(1)  x-\-y — 1=0  and  x-^y — 3  =  o 

(2)  a;  +  3/  +  1  =  o  and  x  —  y  —  3  =  o 

(3)  X  -{-y  —  1=0  and  a:  +  y  +  3  =  o 

(4)  X  -'ry  +  \~o  and  x  +  y  -{-2  =  0. 

Equations  (3)  and  (4)  are  contradictory. 


11. 


a;'  —  2yx  +  8  =  o 
:r'  —  2y'   +  14=0. 


Final  eq.uatiop,  y^  —  8y^  -^^ 9  =  o.     Common  divisor,  yx- 
f  +  3. 

Solutions. 
y=3,  y  =  — 3,  y=  +  v/3i;  y  =  —  y^—i 
x=2,  X  =.—  2,  X  -  4y/ZIY,  *  =  —  4 \/HX 


41 


^{y  —  \)x'+y{y-\-\)x'+{^f-\-y—2)x-\-2y  =  o 


'■■  \ra 


)x'+y{y+i)x  +  3f  —  l  =  o. 


Final  equations,  y'  —  1  =  o.    jDommon  divisor,  {y  —  J ) 
X  +  2y. 

The  value  of  y  =  1  must  be  rejected  (Art.  29,)  since  it  re- 
duces the  common  divisor  to  2. 


Solutions. 
y  =  —  I,         a;  =  —  1, 


ANALYSIS   OF  CURVES. 


1.  We  have  seen  in  Analytical  Geometry,  that  every 
equation  between  two  indeterminates,  may  be  conceived  to 
express  the  relation  between  the  abscissas  and  ordinates  of 
the  curve,  which  this  equation  represents.  By  giving  par- 
ticular values  to  either  of  the  variables,  the  corresponding 
values  of  the  other  may  be  deduced,  and  all  the  points  of  the 
curve  determined.  F(;llowing  the  course  therein  defined, 
we  may  ascertain  whether  or  not  this  curve  is  symmetrical 
with  respect  to  either  or  both  the  co-ordinate  axes ;  we 
may  also  define  its  limits  when  any  exist,  by  determining  the 
points  at  which  the  tangents  are  parallel  to  the  axes,  or  by 
ascertaining  the  existence  and  position  of  its  asymptotes. 

2.  Beyond  this,  however,  the  powers  of  Analytical  Geom- 
etry end,  and  we  are  compelled  to  resort  to  those  means 
which  the  discovery  of  the  science  of  the  Differential  Cal- 
culus has  placed  at  our  command.  By  this  we  may  not  on- 
ly verify  the  results  of  the  geometrical  analysis,  but  we  may 
trace  with  the  most  exact  certainty,  the  course  of  any  curve 
however  irregular,  and  define  its  properties  however  pecu- 
liar. The  sole  difficulty  consists  in  solving  the  algebraic- 
equation  which  defines  the  curve.  If  this  difficulty  be  re- 
moved, we  may  readily  trace  its  course.  For,  suppose  that 
the  equation  of  the  curve  has  been  solved,  and  that  X,  X',- 


44 

X",  &c.  represent  the  roots  of  y,  these  roots  being  functions 
of  x;  the  question  is  at  once  reduced  to  an  examination  of 
the  particular  curves,  which  are  expressed  by  the  separate 
equations 

1/c.X,  y=X',  3/=X",«fec. 

This  examination  will  be  effected  by  giving  to  x  every 
possible  value,  as  well  negative  as  positive,  which  the  func- 
tions X, X',  X",  &c.  admit  of,  without  becoming  imaginary; 
and  the  curves  which  result  will  be  the  different  branches  of 
the  curve  represented  by  the  given  equation.  The  extent 
of  each  of  these  branches  will  depend  upon  the  different  so- 
lutions which  correspond  to  its  particular  equation.  If  any 
of  the  equations 

y  =  X,  y  =  X',  y  =  X", 

exist  for  infinite  values  of  x,  it  follows  that  these  branches 
extend  indefinitely  in  the  direction  of  these  values.  Let  us 
apply  these  principles  to  the  analysis  of  the 


Lemniscate  Curve. 
8.  Take  the  equation 

y*  —  9Ba2y2  _^  looa^x'  —  x'  =  o. 
I'his  being  a  quadratic  equation,  its  solution  is  efTected  by 


45 


the  ordinary  rules  for  such  equations,  and  we  find  the  values 
of  y  to  be 


y  -  ±  \/48a2  ±  V 2304a'  —  lOOa^^z;^  +  x' 
or  putting 

2304a*  —  lOOa^x^  +  x*  =  N, 

the  four  values  of  y  become 


y  =  V  48a2  +  v/N  .  .  (1)      3/  =  ^48a^  —  ^/N  .  .  (2) 


=  — \/48a2+%/N   .  (3)      3/  =  — 'y/48a2_v/N   .  (4) 


It  is  required  now  to  ascertain  each  of  the  curves  which 
these  equations  represent. 

We  see  in  the  first  place,  that  the  values  (3)  and  (4)  only 
diflTer  from  those  of  (1)  and  (2)  in  the  sign,  and  consequently 
must  represent  similar  branches  whose  position  with  respect 
to  the  axis  of  x  alone  differs.  Further,  as  the  quantity  N 
contains  even  powers  of  x  only,  its  value  will  not  be 
changed  by  substituting  a  negative  for  a  positive  value  of  x. 
The  parts  of  the  curve  which  lie  on  the  right  of  the  axis  of 
y,  are  therefore  similar  to  those  which  lie  on  the  left  of  this 
7 


46 

axis.     Hence  the  curve  is  divided  by  the  co-ordinate  axes 
into  four  equal  and  symmetrical  parts. 

4.  Let  us  now  examine  more  particularly  the  values  (1) 
and  (2). 

They  can  only  be  real  so  long  as  the  quantity  N  is  posi- 
tive ;  the  Umit  to  the  real  values  of  y  will  then  be  found  by 
making 

N  =  a:*  —  lOOa^^s  +  2304a*  =  o. 

But  this  equation  can  be  decomposed  into  the  factors 
X  —  6a,  X  +  6a,  x  —  8a,  x  -\-  8a, 

and  the  values  of  y  for  equation  (1)  will  be 

y  =  V    48a2  -j-  v/(.y  _  6a)  {x  +  6a)  {x  —  8a)    {x  +  8a) 


For  any  values  oix  greater  than  6a,  but  less  than  8a,  the 
values  of  y  will  be  imaginary,  since  the  factor  {x  —  8a)  un- 
der this  supposition  is  negative.  No  part  of  the  curve  then  is 
embraced  within  the  Umits 


X  =  Qa,  X  =  8a  ; 

but  for  values  of  x  greater  than  8a,  the  factor  {x  —  Sa) 
becomes  positive,  and  the  values  of  y  always  real. 


47 
The  values  of  y  which  correspond  to  the  three  values  of  a:, 

X  =  0,  X  —  Qa,  X  =  8a, 

will  be  found  from  equation  (1)  to  be 


Equation  (1)  gives  then,  1st,  a  part  DF  (see  Figure  1) 
"which  extends  from  the  point  D,  taken  on  the  axis  AC,  to 
the  point  F,  whose  abscissa  AE  =  6a;  2ndly,  a  part  HX, 
which  beginning  at  the  point  H,  whose,  abscissa  AG  =  8a, 
extends  indefinitely  in  the  angle  BAG. 

5.  Equation  (2),  which,  when  the  factors  of  N  are  intro- 
duced, becomes 


y  =  V    48a2  —  V  (^  _  6a)  [x  +  6a)  {x  —  8a)  {x  +  8a), 

will  in  like  manner  give  imaginary  results  between  the  limits 

a:  =  6a,    a:  =  8a ; 
but  for  the  values 

X  =  0,  X  =  Qcux  =  8a, 


48 


we  get 

y  =  0,  y=  "v^isi?,  y  =  ^^iSo^, 


which  show,  1st,  that  equation  (2)  gives  a  part  AF,  which 
unites  with  the  part  DF  given  by  equation  (1)  at  the  point 
F,  for  which  the  two  ordinates  are  equal ;  2ndly,  beginning 
at  the  point  H,  equation  (2)  gives  a  part  HK,  in  which  y 
decreases  until  VN  =  48a^,  when  it  becomes  zero,  and  cor- 
responds to  the  point  I.  For  N  greater  than  48a^  the  quan- 
tity under  the  radical  becomes  negative,  and  y  imaginary. 
The  Branch  of  the  Curve  corresponding  to  equation  (2) 
does  not  therefore  extend  beyond  the  point  I.  The  abscissa 
of  this  point  is  evidently  determined  by  making  y  —  o'ln 
equation  (2).     We  find 

X  =  ±  o,    X  =  ±  10a. 

The  two  first  values  correspond  to  the  point  A,  the  others 
to  the  points  I  and  I'. 

6.  We  might  continue  this  discussion,  which  is  in  every 
respect  analogous  to  the  general  discussion  of  an  equation  of 
the  second  degree  in  analytical  geometry,  and  ascertain 
whether  this  curve  has  asymptotes ;  but  as  the  differential 
calculus  abridges  this  investigation,  we  will  at  once  apply  it 
to  this  purpose,  and  then  proceed  to  the  determination  of  the 
singular  points  of  the  curve. 

7.  An  examination  of  the  four  values  of  y,  Arts.  3  and  4, 


Fig.  2. 


50 

has  already  shown  that  the  curve  we  are  discussing  has  in 
each  angle  of  the  co-ordinate  axes  an  indefinite  branch.  Let 
us  see  whether  these  branches  have  asymptotes. 

We  know  that  if  any  curve  MX  (Fig.  2)  have  an  asymp- 
tote RS,  the  tangent  MT  approaches  more  and  more  a  co- 
incidence with  the  asymptote  as  the  point  of  tangency  is  re- 
moved from  the  origin.  Under  this  supposition,  the  points 
T  and  D  in  which  the  tangent  intersects  the  axes,  will  con- 
tinually approach  the  points  R  and  E,  in  which  the  asymp- 
totes intersect  the  axes  ;  so  that  AR  and  AE  are  limits  to 
the  values  of  AT  and  AD.  Hence,  to  ascertain  whether  a 
curve  has  asymptotes,  it  is  necessary  to  determine  whether 
the  expressions  AT  and  AD,  which  represent  the  distances 
from  the  origin  to  the  points  in  which  the  tangent  cuts  the 
co-ordinate  axes,  have  limits  for  infinite  values  of  x  and  y. 
If  they  have,  these  limits  being  constructed  will  give  the 
points  D  and  E,  through  which,  if  the  line  RS  be  drawn,  it 
will  be  the  asymptote  sought. 

8.  The  expressions  for  AT  and  AD  may  be  deduced  at 
once  from  the  equation  of  the  tangent  line.  The  equation  of 
the  tangent  line  is 


ax 


^  being  the  tangent  of  the  angle  which  it  makes  with  the 
dx 

axis  of  X.  We  may  now  obtain  the  distances  AT  and  AD, 
by  making  y'  and  x'  separately  equal  to  zero.  By  the  first 
supposition  we  have  


51 


a;'=AT=a;  — y  —^ 


in  which  the  quantity y—>  which  is  the  expression  for  the 
dy 

subtangent  PT,  is  taken  negatively,  since  it  is  counted  in  an 
opposite  direction  from  the  abscissa  x'.  Making  now  x'  =  o, 
we  have 


v'  =  AD  =  V  —  X  — ^. 
^  ^  dx 


9.  To  ascertain  whether  the  given  curve  has  asymptotes, 

we  must  substitute  the  values  of  -^    and   — ,  deduced  from 

dx  dy 

the  equation  of  the  curve,  in  the  expressions  for  AT  and 
AD,  and  see  what  these  expressions  become  when  x  and  y 
are  infinite.  We  find, the  first  differential  co-efficient  of  the 
given  equation  after  dividing  by  4,  to  be 


dx    _  y'  —  ^Sa^y 
dy         x^  —  bOa^x 

Multiplying  this  value  by  y,  and  subtracting  the  product 
from  a;,  we  have  after  reducing. 


^  Jy  x'  -  SOa^x 

By  a  simple  transposition  of  the  terms  of  the   fraction, 


52 

which  forms  the  value  of  — ,  we  deduce  that  of  -^,  and 

dy  dx 

we  have 


AD  =  V  — a:  ^  =  !/^-48gy-a:^+  bQa^x\ 
^  dx  y'  —  4Sa^y 


Putting  in  these  expressions  the  value  of  y*,  they  become 


x"  —  50a^x 


y^  —  48a^y 


These  values  of  AT  and  AD,  continually  diminish  as  x 
and  y  increase,  and  when  x  and  y  equal  ±  infinity,  they  be- 
come zero.  We  conclude,  then,  that  the  curve  has  two 
asymptotes,  which  pass  through  the  origin  of  co-ordinates. 

Their  angle  is  determined  by  seeing  what  -^  becomes  when 

X  and  y  equal  ±  infinity.     We  have 

dy  _  x^  —  50a^x 
dx      y^  —  48a^y 

when  X  and  y  are  infinite,  the  first  powers  of  y  and  x  may  be 
neglected,  and  we  have 


53 


ax 


which  shows  that  one  of  the  asymptotes  makes  an  angle 
with  the  axis  of  x  of  45°  ;  the  other  an  angle  of  45°  +  90° 
=  135°. 

10.  Let  us  now  examine  the  singular  points  of  the  curve. 
We  find  the  first  differential  co-efficient  to  be 


dy  _x^  —  50a^a: 
dx      y'  —  48a^i/' 


To  determine  the  points  at  which  the  tangent  is  parallel 

to  the  axis  of  x,  make  -^  ±  o.    We  find 
dx 


x^  —  SOa^x  =  Of 
which  gives 


X  =  o    a;  =  +  VSOa^      x  =  —  VdOa^ 

The  value  of  x—  o,  when  substituted  in  the  given  equa- 
tion, gives 


y  =  ±0      y  =  ±  y/9Qa^. 

But  when  x  =■  o  and  a:  =  o,  we  have 
8 


54 

dy  _x^  —  50a'ar  _  o 
dx      y^  —  48a^y      o 

■which  indicates  a  multiple  point  at  the  origin  of  co-ordinates, 
(see  Boucharlat's  Differential  Calculus,  Art.  138.) 

To  determine  the  value  of  -^,  we  must  pass  to  the  second 

dx 

differential  equation,  which  becomes,  when  x  -  o  and  y  =  o, 
—  48a%2  +  50a^dx^  =  o. 

Hence 


I=*v1 


^50 
dx      ~  V    4q' 


It  follows  from  these  values,  that  at  the  point  A,  the  curve 
is  touched  by  two  straight  lines,  which  make  angles  with 
the  axis  of  x,  the  tangents  of  which  are 


+  v^^,  and   -v/-5 
^48  ^48 


the  point  A  is  therefore  a  multiple  point. 


11.  The  values  of  y  =  ±  V9Ga^  correspond  to  the  points 
D  and  D',  at  which  the  tangent  is  parallel  to  the  axis  of  x. 

12.  If  we  deduce  the  3d  differential  equation,  we  find  by- 
making  X  =  o,y  =^  0, 

—  48a2  dy  d^y  —  QGa'dy  d^y  =  o, 


55 
or 

-f  144a^  dy  d}y  =o; 

from  which  we  conclude  that 


The  second  differential  co-efficient  being  zero,  let  us  find 
the  third  differential  co-efficient  from  the  fourth  differential 
equation.     This  equation,  when  we  make  x  —  o,y  =  o,  and 

—^  =  0,  reduces  to 


—  4.48a2  dy  d'y  -f  Qdy'  —  6dx*  =  o 
from  which  we  deduce 


dx  dx'      dx* 


Hence 


dx' 


\48/ 


^    48* 


56 

when  -^  is  replaced  by  its  value  ±  v/ — 
dx  ^    48* 

But  we  have  found  (Boucharlat,  Art.  123),  for  the  dis- 
tance between  the  tangent  and  the  curve, 


dx'\.2         dxn.2.2 


When  we  make  in  this  expression 

dx" 
and 


>         148/ 


r50\« 

dx' 


^    48 


it  becomes 


_  .  Vis/  h' 


—  1 

5  =  ±    ^'^^^      V    ^^^ +  &C. 

7^      1-2.3 


48 


which  shows  that  the  branch  of  the   curve   touched  by  the 
straight  line  AL,  which  corresponds  to  the  positive  value  of 


57 

-^,  is  above  the  straight  line  on  the  side  of  the  positive  ab- 
scissas ;  and  below  it,  on  the  side  of  the  negative  abscissas. 
The  reverse  takes  w^ith  respect  to  the  tangent  AL'.  Hence 
each  branch  of  the  curve  undergoes  an  inflexion  at  the 
point  A. 

13.  If  in  the  2nd  differential  equation 

3fdy^+y3d^y—'^a^di/—48ah/(Py4-b0aPdx^-\-50a^xiPx—33^dx^—x3(Pz  =  o, 


we  make  x  =  o,  and  y  =  ±  v^QGa'  and  -7-  =  o,  we  have 


^  =  —  (±  V96ay  —  48a2  x  ±  ^/96a^ 


which  gives  a  negative  value  for  — ^  for  the  value  of  y,  cor- 

dx^ 

responding  to  the  point  D,  and  a  positive  value  for  D',  which 
shows  that  at  D,  the  ordinate  is  a  maximum,  while  at  D'  it 
is  a  mitiimum.  The  ordinate  at  D'  is  regarded  as  a  mini- 
mum, because  every  increment  to  a  negative  ordinate  is 
equivalent  to  a  decrement  with  respect  to  positive  ordinates. 

14.  To  find  the  points  at  which  the  tangents  are  perpen- 
dicular to  the  axes  of  a;,  we  must  put  ^  =    6,   or  what  is 

dx 

equivalent  to  it,  place  the  denominator  of  its  value  equal  to 
zero.     This  gives 


58 
y"  —  48a'y  =  o, 


from  which  we  obtain 


y  =  o  y  =  ±  s/48a2. 

The  first  value  substituted  in  the  equation  of  the  curve 
gives 

lOOaV— a;*  =  o. 
Hence 

X  =  ±0  X  r=  ±  10a. 

The  roots  x  =  ±  o  indicate  the  multiple  point  A,  the 
two  others  belong  to  the  points  I  and  I'.  The  values 
y  =  ±  V48a''  correspond  to  the  points  whose  abscissas  are 

X  =  ±6a  X  =  ±  8a. 

One  of  these  results  makes  known  the  points  F,  F',  the 
other  the  points  H,  H'.  At  each  of  these  points  the  tangent 
is  perpendicular  to  the  axis  of  a:,  and  as  there  are  no  points 
of  the  curve  between  these  limits,  we  can  readily  see  in 
what  direction  the  curve  is  turned  towards  its  tangents. 


59 

15.  We  might  continue  this  discussion,  and  by  analogous 
means  ascertain  the  singular  points  belonging  to  the  branches 
represented  by  the  equations  (3)  and  (4) ;  but  as  we  have 
already  seen  that  these  curves  are  in  every  respect  identical 
with  those  just  discussed,  the  number  and  character  of  the 
singular  points  may  be  regarded  as  known. 


J.  p.  Wright,  Printer,  18  New  Screet,  N.  T. 


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